Method for gradually deforming sequential simulations of a heterogeneous environment such as an underground zone

ABSTRACT

A method of gradual deformation of representations or realizations, generated by sequential simulation, not limited to a Gaussian stochastic model of a physical quantity z in a meshed heterogeneous medium, in order to adjust the model to a set of data relative to the structure or the state of the medium which are collected by previous measurements and observations. The method comprises applying a stochastic model gradual deformation algorithm to a Gaussian vector with N mutually independent variables which is connected to a uniform vector with N mutually independent uniform variables by a Gaussian distribution function so as to define realizations of the uniform vector, and using these realizations to generate representations of the physical quantity z that are adjusted to the data.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method of gradual deformation ofrepresentations or realizations, generated by sequential simulation, ofa model of a heterogeneous medium which is not limited to a Gaussianstochastic model, based on a gradual deformation algorithm of Gaussianstochastic models.

2. Description of the Prior Art

In French patent application 98/09,018 a method is described whichgradually deforms a stochastic (Gaussian type or similar) model of aheterogeneous medium such as an underground zone, constrained by a setof parameters relative to the structure of the medium. This methodcomprises drawing a number p (p=2. for example) of realizations (orrepresentations) independent of the model or of at least part of theselected model of the medium from all the possible realizations and oneor more iterative stages of gradual deformation of the model byperforming one or more successive linear combinations of p independentinitial realizations and then composite realizations are successivelyobtained possibly with new draws, etc., the coefficients of thiscombination being such that the sum of their squares is 1.

Gaussian or similar models are well-suited for modelling continuousquantity fields but are therefore ill-suited for modelling zones crossedby fracture networks or channel systems for example.

The most commonly used geostatistical simulation algorithms are thosereferred to as sequential simulation algorithms. Although sequentialsimulation algorithms are particularly well-suited for simulation ofGaussian models, these algorithms are not in principle limited to thistype of model.

A geostatistical representation of an underground zone is formed forexample by subdividing thereof by a network with N meshes and bydetermining a random vector with N dimensions Z=(Z₁, Z₂, . . . Z_(N))best corresponding to measurements or observations obtained on the zone.As shown for example by Johnson, M. E.; in “Multivariate StatisticalSimulation”; Wiley & Sons, New York, 1987, this approach reduces theproblem of the creation of an N-dimensional vector to a series of None-dimensional problems. Such a random vector is neither necessarilymulti-Gaussian nor stationary. Sequential simulation of Z first involvesthe definition of an order according to which the N elements (Z₁, Z₂, .. . Z_(N)) of vector Z are generated one after the other. Apart from anyparticular case, it is assumed that the N elements of Z are generated insequence from Z₁ to Z_(N). To determine a value of each element Z₁,(i=1, . . . , N), the following operations have to be carried out:

a) building the distribution of Z_(i) conditioned by (Z₁, Z₂. . .Z_(i-1)) F_(c)(Z_(i))=P(Z_(i)≦/Z₁, Z₂, . . . Z_(i-1)); and

b) determining a value of Zi from distribution F_(c) (Z_(i)).

In geostatistical practice, sequential simulation is frequently used togenerate multi-Gaussian vectors and non-Gaussian indicator vectors. Themain function of sequential simulation is to determine conditionaldistributions F_(c) (Z_(i)) (i=1, . . . , N). Algorithms and softwaresfor estimating these distributions are for example described in:

Deutsch, C. V. et al. “GSLIB (Geostatistical Software Library) andUseres Guide”; Oxford University Press, New York, Oxford 1992.

Conceming determining the values from distribution F_(c) (Z_(i)), therealso is a wide set of known algorithms.

The inverse distribution method is considered by means of which arealization of Z_(i); _(i)=F_(c) ⁻¹(u_(l)) is obtained, where u₁ istaken from a uniform distribution between 0 and 1. A realization ofvector Z therefore corresponds to a realization of vector U whoseelements U₁, U₂, . . . , U_(N), are mutually independent and evenlydistributed between 0 and 1.

It can be seen that a sequential simulation is an operation S whichconverts a uniform vector U=(U₁, U₂, . . . U_(N)) to a structured vectorZ=(Z₁, Z₂, . . . , Z_(N)):

Z=S(U)  (1).

The problem of the constraint of a vector Z to various types of data canbe solved by constraining conditional distributions F_(c)(Z_(i)) (i=1, .. . , N) and/or uniform vector U=(U₁, U₂, . . . , U_(N)).

Recent work on the sequential algorithm was focused on improving theestimation of conditional distributions F_(c) (z_(i)) by geologic dataand seismic data integration. An article by Zhu, H. et al: “Formattingand Integrating Soft Data: Stochastic imaging via the Markov-BayesAlgorithm” in Soares, A., Ed. GeostaUstics Troia 92, vol.l: KiuwerAcad.Pubi., Dordrecht, The Netherlands, pp.1-12, 1993 is an example.

However, this approach cannot be extended to integration of non-lineardata such as pressures from well tests and production records, unless asevere linearization is imposed. Furthermore, since any combination ofuniform vectors U does not give a uniform vector, the method for gradualdeformation of a stochastic model developed in the aforementioned patentapplication cannot be directly applied within the scope of thesequential technique described above.

The method according to the invention thus allows making the twoapproaches compatible, that is to extend the formalism developed in theaforementioned patent application to gradual deformation ofrealizations, generated by sequential simulation, of a model which isnot limited to a Gaussian stochastic model.

SUMMARY OF THE INVENTION

The method allows gradual deformation of a representation orrealization, generated by sequential simulation, of a model which is notlimited to a Gaussian stochastic model of a physical quantity z in aheterogeneous medium such as an underground zone, in order to constrainthe model to a set of data collected in the medium by previousmeasurements and observations relative to the state or the structurethereof.

The method of the invention comprises applying an algorithm of gradualdeformation of a stochastic model to a Gaussian vector (Y) having anumber N of mutually independent variables that is connected to auniform vector (U) with N mutually independent uniform variables by aGaussian distribution function (G), so as to define a chain ofrealizations u(t) of vector (U), and using these realizations u(t) togenerate realizations z(t) of this physical quantity that are adjustedin relation to the (non-linear) data.

According to a first embodiment, the chain of realizations u(t) ofuniform vector (U) is defined from a linear combination of realizationsof Gaussian vector (Y) whose combination coefficients are such that thesum of their squares is one.

According to another embodiment, gradual deformation of a number n ofparts of the model representative of the heterogeneous model isperformed while preserving the continuity between these n parts of themodel by subdividing uniform vector (U) into a number n of mutuallyindependent subvectors.

The method of the invention finds applications in modeling undergroundzones which generates representations showing how a certain physicalquantity is distributed in an underground zone (permeability z forexample) and which is compatible in the best manner with observed ormeasured data: geologic data, seismic records, measurements obtained inwells, notably measurements of the variation with time of the pressureand of the flow rate of fluids from a reservoir, etc.

BRIEF DESCRIPTION OF THE DRAWINGS

Other features and advantages of the method according to the inventionwill be clear from reading the description hereafter of a non limitativeexample, with reference to the accompanying drawings wherein:

FIG. 1 shows the medial layer of a realization of a facies model used asa reference, generated by sequential simulation of indicatrices,

FIG. 2 shows the variation with time of the pressure obtained in a welltest for the reference model,

FIGS. 3A to 3E respectively show five initial realizations of the mediallayer of a reservoir zone, constrained only by the facies along thewell,

FIGS. 4A to 4E respectively show, for these five realizations, thebottomhole pressure curves in the reference model compared with thosecorresponding to the initial models,

FIGS. 5A to 5E respectively show five realizations of the medial layerof the facies model conditioned to the facies along the well andadjusted in relation to the pressure curve obtained by well tests,

FIGS. 6A to 6E respectively show, for the five realizations, thebottomhole pressure curves in the reference model compared with thosecorresponding to the adjusted models,

FIGS. 7A to 7E respectively show how the objective functionsrespectively corresponding to these five examples vary with the numberof iterations,

FIGS. 8A to 8E show the gradual deformations generated by an anisotropycoefficient change on a three-facies model generated by sequentialsimulation of indicatrices, and

FIGS. 9A to 9E show the local gradual deformations of a three-faciesmodel, generated by sequential simulation of indicatrices.

DETAILED DESCRIPTION OF THE INVENTION

A study zone is considered that is subdivided by an N-mesh grid.Realizations or representations of a stochastic model of a certainphysical quantity z representing for example the permeability of theformations in the zone are attempted to be made. The model that issought must adjust to data obtained by measurements or observations at acertain

Adjustment of a stochastic model to non-linear data can be expressed asan optimization problem. The quantity f^(obs=) (f₁ ^(obs), f₂ ^(obs), f₃^(obs). . . f_(p) ^(obs)) designates the vector of the nonlinear dataobserved or measured in the studied medium (the reservoir zone), and thequantity f=(f₁, f₂, f₃ . . . f_(p)) is the corresponding vector of theresponses of the stochastic model of the permeability Z=(Z₁, Z₂, . . . ,Z_(N),). The problem of constraining the stochastic model of Z byobservations consists in generating a realization z of Z which reducesto a rather low value an objective function that is defined as the sumof the weighted rms errors of the responses of the model in relation tothe observations or measurements in the reservoir zone, i.e.:$O = {\frac{1}{2}\quad {\sum\limits_{i = 1}^{p}\quad {\omega_{i}\quad \left( {f_{1} - f_{1}^{obs}} \right)^{2}}}}$

where ω_(i), represents th weight assigned to respons f_(i) Functionsf_(i) (i=1, 2, . . . , p) and objective function O are functions ofvector Z. This presents an optimization problem of dimension N.

In order to extend the formalism developed in the aforementioned patentapplication to the gradual deformation of realizations generated by, butnot necessarily limited to Gaussian sequential simulation, a startingpoint is from a Gaussian vector with N variables Y_(i), with i=1, 2, . .. , N, mutually independent, of zero mean and of variance equal to 1,and N mutually independent uniform variables U_(i), U₂, U₃, . . . U_(N)are defined by:

U_(i)=G(Y_(i))∀i=1, 2, . . . , N

where G represents the standardized Gaussian distribution function.

Assuming this to be the case, the gradual deformation algorithmdeveloped within a Gaussian frame is applied to the Gaussian vectorY=(Y₁, Y₂, . . . , Y_(N)) in order to build a continuous chain ofrealizations of uniform vector U (U₁, U₂, . . . , U_(N)). Given twoindependent realizations Y_(A) and Y_(b) of Y, the chain of realizationsu(t) of vector U obtained with the following relation is defined:

u(t)=G(y _(a), cos t+y_(b), sin t)  (2).

For each t, u(t) is a realization of vector U. A vector z(t) which is,for each t, a realization of random vector Z is then obtained bysampling of the conditional distribution F_(c) (z_(i)) (i=1, 2, . . . ,N) using the elements of vector u(t). Parameter t can consequently beadjusted as in the Gaussian case so as to adjust z(t) to non-lineardata. This procedure is iterated until satisfactory adjustment isobtained.

Adjustment of a Facles Model to Pressure Data Obtained by Means of WellTests

In order to illustrate application of the stochastic optimization methoddefined above, adjusting a stochastic reservoir model to pressure dataobtained by means of well tests is attempted. Building of the reservoirmodel is derived from a real oil formation comprising three types offacies: two reservoir facies of good quality (facies 1 and 2) and areservoir facies of very bad quality (facies 3). Table 1 defines thepetrophysical properties of the three facies:

K_(x) (md) K_(y) (md) K_(z) (md) φ (%) c_(t) (10⁻⁵ bar⁻¹) Facies 1 10 1010 17 2.1857 Facies 2 1 1 1 14 2.0003 Facies 3 0.1 0.1 0.001 9 1.8148

In order to represent the specific facies distribution of the oilformation, a binary realization is first generated to represent fades 3and its complement. Then, in the complementary part of facies 3, anotherbinary realization independent of the first one is generated torepresent facies 1 and 2. The formation is discretized by means of aregular grid pattern of 60×59×15 blocks 15 m×15 m×1.5m in size. Anexponential variogram model is used to estimate the conditionaldistributions. The main anisotropy direction is diagonal in relation tothe grid pattern. The ranges of the variogram of facies 3 in the threeanisotropy directions are 300 m, 80 m and 3 m respectively. The rangesof the variogram of facies 1 and 2 in the three anisotropy directionsare 150 m, 40 m and 1.5 m respectively. The proportions of facies 1, 2,3 are 6%, 16% and 78% respectively.

A well test has been carried out by means of a finite-difference welltest, simulator as described by: Blanc, G. et al: “BuildingGeostatistical Models Constrained by Dynamic Data—A PosterioriConstraints” in SPE 35478, Proc. NPF/SPE European 3D Reservoir ModellingConference, Stavenger, Norway, 1996.

The medial layer of a realization used as the reference model for thisvalidation can be seen in FIG. 1. The section of the well that has beendrilled runs horizontally through the medial layer of the reservoirmodel along axis x. The diameter of the well is 7.85 cm, the capacity ofthe well is zero and the skin factors of facies 1, 2 and 3 are 0, 3 and50 respectively. The synthetic well test lasts for 240 days with aconstant flow rate of 5 m³/day so as to investigate nearly the entireoil field. FIG. 2 shows the pressure variation with time.

The objective was to build realizations of the facies model constrainedby the facies encountered along the well and by the pressure curveobtained during well testing. The objective function is defined as thesum of the rms differences between the pressure responses of thereference model and the pressure responses of the realization. Since thedynamic behavior of the reservoir model is mainly controlled by thecontrast between the reservoir facies of good and bad quality, thebinary realization used to generate facies 1 and 2 has been fixed firstand only the binary realization used to generate facies 3 has beendeformed for pressure data adjustment.

The pressure responses resulting from the well tests for the fiverealizations of FIGS. 3A to 3E are different from that of the referencemodel, as shown in FIGS. 4A to 4E. Starting respectively from these 5independent realizations, by using the iterative adjustment methodabove, after several iterations, five adjusted are obtained realizations(FIGS. 5A to SE) for which the corresponding pressure curves are totallyin accordance with those of the reference model, as shown in FIGS. 6A to6E.

Gradual Deformation in Relation to the Structural Parameters

In many cases, sufficient data for deducing the structural parameters ofthe stochastic model: mean, variance, covariance function, etc., are notavailable. These structural parameters are often given in terms of apriori intervals or distributions. If their values are wrong, it isuseless to seek a realization adjusted to non-linear data, It istherefore essential for applications to be able to perform a gradualdeformation of a realization with simultaneous modification of randomnumbers and structural parameters. The sequential simulation algorithmdefined by equation (1) makes possible changing, simultaneously orseparately, a structural operator S and a uniform vector U. FIGS. 8A to8E show the gradual deformations obtained for a fixed realization ofuniform vector U when the anisotropy coefficient is changed.

Local or Regionalized Gradual Deformation

When the observations are spread out over different zones of a studiedformation, an adjustment using global deformation would be ineffectivebecause the improvement obtained in a zone could deteriorate theimprovement in another zone. It is therefore preferable to apply gradualdeformations zone by zone. Consider a subdivision of vector U into acertain number n of mutually independent subvectors U¹, U², . . . ,U^(n), which allows performing their gradual deformation individually.Separate application of the gradual deformation algorithm to eachsubvector U¹, U². . . , U^(n) allows obtaining a function of dimension nof uniform vector U:${U\quad \left( {t_{1},t_{2},\ldots \quad,t_{n}} \right)} = {\begin{bmatrix}{U^{1}\quad \left( t_{1} \right)} \\{U^{2}\quad \left( t_{2} \right)} \\\vdots \\{U^{n}\quad \left( t_{n} \right)}\end{bmatrix} = \begin{bmatrix}{G\quad \left( {{Y_{a}^{1}\quad \cos \quad t_{1}} + {Y_{b}^{1}\quad \sin \quad t_{1}}} \right)} \\{G\quad \left( {{Y_{a}^{2}\quad \cos \quad t_{2}} + {Y_{b}^{2}\quad \sin \quad t_{2}}} \right)} \\\vdots \\{G\quad \left( {{Y_{a}^{n}\quad \cos \quad t_{n}} + {Y_{b}^{n}\quad \sin \quad t_{n}}} \right)}\end{bmatrix}}$

where Y^(i) _(a) and Y^(i) _(b) for any i=1, 2, . . . , n, areindependent Gaussian subvectors. For a set of realizations of V^(i)_(a), and v^(i) _(b), a problem of optimization of n parameters t₁, t₂,. . . , t_(n) is solved to obtain a realization that improves ormaintains the data adjustment This procedure can be iterated untilsatisfactory adjustment is obtained.

Gradual local deformations thus allow significant improvement of theadjustment speed in all the cases where measurements or observations arespread out over different zones of the medium.

The effect of this gradual local deformation on the three-facies modelof FIGS. 9A to 9E can be clearly seen therein where only the enclosedleft lower part is affected.

The method according to the invention can be readily generalized togradual deformation of a representation or realization of any stochasticmodel since generation of a realization of such a stochastic modelalways comes down to generation of uniform numbers.

What is claimed is:
 1. A method for forming a representation generatedby sequential simulation, of a stochastic model, which is not limited toa Gaussian stochastic model, of a physical quantity z in a heterogeneousmedium, in order to constrain the stochastic model to data collected inthe heterogeneous medium by means of previous measurements andobservations, relative to a state or the structure of the heterogeneousmedium, comprising applying a stochastic model gradual deformationalgorithm to a Gaussian vector (Y) with N mutually independent variablesthat is connected to a uniform vector U with N mutually independentuniform variables by a Gaussian distribution function (G), so as to forma chain of realizations u(t) of vector U, and using the chain ofrealizations u(t) to generate realizations z(t) of the physical quantitythat are adjusted to the data with.
 2. A method as claimed in claim 1,wherein: the chain of realizations u(t) of the vector (U) is definedfrom a linear combination of realizations of the Gaussian vector (Y)comprising combination coefficients with a sum of squares of thecoefficients being one.
 3. A method as claimed in claim 1, comprising:gradually deforming the stochastic model, which is not limited to aGaussian stochastic model representative of the heterogeneous medium,simultaneously in relation to the structural parameters and to randomnumbers.
 4. A method as claimed in claim 2, comprising graduallydeforming the stochastic model, which is not limited to a Gaussianstochastic model representative of the heterogeneous medium,simultaneously in relation to the structural parameters and to randomnumbers.
 5. A method as claimed in claim 1, comprising: performing aseparate gradual deformation of a number n of parts of the stochasticmodel, which is not limited to a Gaussian stochastic modelrepresentative of the heterogeneous medium, while preserving continuitybetween the parts of the stochastic model, not limited to a Gaussianstochastic model representative of the heterogeneous medium bysubdividing the uniform vector U into n mutually independent subvectors.6. A method as claimed in claim 2, comprising: performing a separategradual deformation of a number n of parts of the stochastic model,which is not limited to a Gaussian stochastic model representative ofthe heterogeneous medium, while preserving continuity between the partsof the stochastic model, not limited to a Gaussian stochastic modelrepresentative of the heterogeneous medium by subdividing the uniformvector U into n mutually independent subvectors.
 7. A method as claimedin claim 3, comprising: performing a separate gradual deformation of anumber n of parts of the stochastic model, which is not limited to aGaussian stochastic model representative of the heterogeneous medium,while preserving continuity between the parts of the stochastic model,not limited to a Gaussian stochastic model representative of theheterogeneous medium by subdividing the uniform vector U into n mutuallyindependent subvectors.
 8. A method as claimed in claim 4, comprising:performing a separate gradual deformation of a number n of parts of thestochastic model, which is not limited to a Gaussian stochastic modelrepresentative of the heterogeneous medium, while preserving continuitybetween the parts of the stochastic model, not limited to a Gaussianstochastic model representative of the heterogeneous medium bysubdividing the uniform vector U into n mutually independent subvectors.9. A method in accordance with claim 1, wherein: the heterogeneousmedium is an underground zone.
 10. A method in accordance with claim 2,wherein: the heterogeneous medium is an underground zone.
 11. A methodin accordance with claim 3, wherein: the heterogeneous medium is anunderground zone.
 12. A method in accordance With claim 4, wherein: theheterogeneous medium is an underground zone.
 13. A method in accordancewith claim 5 wherein: the heterogeneous medium is an underground zone.14. A method in accordance with claim 6, wherein: the heterogeneousmedium is an underground zone.
 15. A method in accordance with claim 7,wherein: the heterogeneous medium is an underground zone.
 16. A methodin accordance with claim 8, wherein: the heterogeneous medium is anunderground zone.
 17. A method for forming a representation generated bysequential simulation, of a stochastic model, not limited to a Gaussianstochastic model, of a physical quantity in a heterogeneous undergroundformation, which is defined by a set of data collected in theunderground formation by means of previous measurements andobservations, comprising: a) applying a stochastic model gradualdeformation algorithm to a Gaussian vector with mutually independentvariables that is connected to a uniform vector with mutuallyindependent uniform variables by a Gaussian distribution function, so asto form a chain of realizations u of the Gaussian vector; b) using thechain of realizations u to generate representations of the physicalquantity; c) iteratively modifying an objective function that measuresmisfit between data corresponding to representations of the stochasticmodel representing the physical quantity, and the data collected in theunderground formation until obtaining a representation of theunderground formation substantially fitting the data with.
 18. A methodas claimed in claim 17, wherein: a chain of realizations u of theuniform vector is defined from a linear combination of realizations ofthe Gaussian vector with the combination coefficients such that a sum ofsquares of the coefficients is one.
 19. A method as claimed in claim 17,comprising: gradually deforming the stochastic model representative ofthe underground formation with simultaneously modifying structuralparameters of the stochastic model and random numbers.
 20. A method asclaimed in claim 18, comprising: gradually deforming the stochasticmodel representative of the underground formation with simultaneouslymodifying structural parameters of the stochastic model and randomnumbers.
 21. A method as claimed in claim 17, comprising: performing aseparate gradual deformation of a number n of parts of the stochasticmodel representative of the heterogeneous medium while preservingcontinuity between the number of parts of the stochastic model bysubdividing the uniform vector into n mutually independent subvectors.22. A method as claimed in claim 18, comprising: performing a separategradual deformation of a number n of parts of the stochastic modelrepresentative of the heterogeneous medium while preserving continuitybetween the number of parts of the stochastic model by subdividing theuniform vector into n mutually independent subvectors.
 23. A method asclaimed in claim 19, comprising: performing a separate gradualdeformation of a number n of parts of the stochastic modelrepresentative of the heterogeneous medium while preserving continuitybetween the number of parts of the stochastic model by subdividing theuniform vector into n mutually independent subvectors.
 24. A method asclaimed in claim 1 wherein: the physical quantity is permeability of theunderground formation.
 25. A method as claimed in claim 1 wherein: themeasurements comprise geologic data.
 26. A method as claimed in claim 1wherein: the measurements comprise seismic data.
 27. A method as claimedin claim 1 wherein: the measurements from wells in the heterogeneousmedium comprise seismic data.
 28. A method as claimed in claim 27wherein: the measurements obtained from wells comprise time variation ofpressure and flow rate of fluid from a reservoir.
 29. A method asclaimed in claim 17 wherein: the physical quantity is permeability ofthe underground formation.
 30. A method as claimed in claim 17 wherein:the measurements comprise geologic data.
 31. A method as claimed inclaim 17 wherein: the measurements comprise seismic data.
 32. A methodas claimed in claim 17 wherein: the measurements from wells in theheterogeneous medium comprise seismic data.
 33. A method as claimed inclaim 32 wherein: the measurements obtained from wells comprise timevariation of pressure and flow rate of fluid from a reservoir.